Bi-phase codes are widely used for pulse compression in radar systems. Barker codes are especially preferred in these applications because it achieves the best possible MSR. It has been proved that there is no Barker code of odd length greater than 13. On the other hand, even though there is no rigorous proof about the non-existence of Barker codes of even length greater than 4, published research based on simulations show that they do not exist for lengths up to a few thousands.
Since Barker codes are short, additional means to enhance sidelobe suppression are needed. The code of length 13 achieves a MSR of only 22.28 dB which is much less than the 30 dB required in most radar applications. A peak sidelobe from a strong target echo can sometimes weaken or even completely mask the mainlobe of a smaller target echo. This motivates the need for sidelobe suppression filters or mismatched filters at the receiver. Mis-matched filters have been a subject of research in the literature for a long time. These filters achieve improved mainlobe to sidelobe ratio at the cost of some deterioration in the signal to noise ratio.
Prior research in the area of sidelobe suppression filters can be broadly classified into two general methods. In the first method, a matched filter is first used to perform the pulse compression correlation. A mismatched filter is then used in cascade with the matched filter to suppress the sidelobes. Rihaczek and Golden introduced the R-G filters which follow this methodology and have been a subject of active research ever since their introduction in 1971. The R-G filters are presented in the publication: A. W. Rihaczek and R. M. Golden, Range Sidelobe Suppression for Barker Codes, IEEE Transactions on Aerospace and Electronic Systems, Vol AES-7, No. 6, November 1971, pp 1087-1092. The R-G filters were further improved by Hua and Oskman who proposed a new algorithm to optimize the filter coefficients of the R-G filters. This work is presented in the publication: Chen Xiao Hua and Juhani Oskman, A New Algorithm to Optimize Barker Code Sidelobe and Suppression Filter, IEEE Transactions on Aerospace and Electronic Systems, Vol AES-26, No. 4, July 1990, pp 673-677. It is also described in the U.S. Pat. No. 5,070,337 by the same authors.
The second method involves directly designing the mismatched filter for the Barker code without first passing it through a matched filter. These filters have been designed using the Least Mean Square (LMS) and Linear Programming (LP) algorithms. The LMS approach can be found in the publication: M. H. Ackroyd and F. Ghani, Optimum Mismatched Filters for Sidelobe Suppression, IEEE Transactions on Aerospace and Electronic Systems, Vol AES-9, No. 2, March 1973, pp 214-218. The LP approach was used by Zoraster to optimize the coefficients of the filter designed to minimize the peak range sidelobe of the Barker coded waveform. The LP filters were found to be more effective in peak sidelobe suppression. This technique was introduced in the publication: S. Zoraster, Minimum Peak Range Sidelobe Filters for Binary Phase Coded Waveforms, IEEE Transactions on Aerospace and Electronic Systems, Vol AES-16, No. 1, January 1980, pp 112-115.
In this work, we design the mismatched filter in cascade with a matched filter. The mismatched filter is based on an implementation of a truncated and modified multiplicative inverse of the autocorrelation function of the Barker code used. The inverse filter should not be implemented directly due to its instability and unacceptable delay. Rihaczek and Golden approximated the inverse filter by expanding the transfer function as a Fourier series and then taking finite terms from the expansion. The proposed filter is fundamentally different from this approach because we apply a multiplicative expansion to approximate the transfer function of the inverse filter. In: A. T. Fam, MFIR Filters: Properties and applications, IEEE Trans. Acoust., Speech and Signal Processing, Vol. ASSP-29, No. 6, pp. 1128-1136, December 1981, a multiplicative expansion of IIR filters is used to generate computationally efficient multiplicative FIR filters that approximate their performance. This multiplicative expansion is applied in this work to the non-causal autocorrelation function. The key point of this invention is that following the truncation of this multiplicative expansion, a set of judiciously chosen parameters are added to the model and optimized to achieve the superior performance of the proposed mismatched filter.